# Properties

 Label 1850.2.a.be.1.4 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.1791440.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 9x^{3} + 13x - 4$$ x^5 - 9*x^3 + 13*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.53175$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} +4.67211 q^{7} +1.00000 q^{8} -0.653743 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} +4.67211 q^{7} +1.00000 q^{8} -0.653743 q^{9} +0.0451320 q^{11} +1.53175 q^{12} +5.26514 q^{13} +4.67211 q^{14} +1.00000 q^{16} +3.60861 q^{17} -0.653743 q^{18} -6.22000 q^{19} +7.15650 q^{21} +0.0451320 q^{22} -2.20164 q^{23} +1.53175 q^{24} +5.26514 q^{26} -5.59662 q^{27} +4.67211 q^{28} -4.20386 q^{29} -3.01051 q^{31} +1.00000 q^{32} +0.0691309 q^{33} +3.60861 q^{34} -0.653743 q^{36} +1.00000 q^{37} -6.22000 q^{38} +8.06487 q^{39} -7.38299 q^{41} +7.15650 q^{42} -5.54789 q^{43} +0.0451320 q^{44} -2.20164 q^{46} -4.28072 q^{47} +1.53175 q^{48} +14.8286 q^{49} +5.52749 q^{51} +5.26514 q^{52} +6.10215 q^{53} -5.59662 q^{54} +4.67211 q^{56} -9.52749 q^{57} -4.20386 q^{58} +13.5275 q^{59} -4.27299 q^{61} -3.01051 q^{62} -3.05436 q^{63} +1.00000 q^{64} +0.0691309 q^{66} +12.3751 q^{67} +3.60861 q^{68} -3.37235 q^{69} -4.49354 q^{71} -0.653743 q^{72} +7.03811 q^{73} +1.00000 q^{74} -6.22000 q^{76} +0.210861 q^{77} +8.06487 q^{78} -8.72499 q^{79} -6.61139 q^{81} -7.38299 q^{82} -0.880231 q^{83} +7.15650 q^{84} -5.54789 q^{86} -6.43926 q^{87} +0.0451320 q^{88} +9.97602 q^{89} +24.5993 q^{91} -2.20164 q^{92} -4.61135 q^{93} -4.28072 q^{94} +1.53175 q^{96} +0.240408 q^{97} +14.8286 q^{98} -0.0295047 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9}+O(q^{10})$$ 5 * q + 5 * q^2 + 5 * q^4 - q^7 + 5 * q^8 + 3 * q^9 $$5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9} + 3 q^{11} - 6 q^{13} - q^{14} + 5 q^{16} + 9 q^{17} + 3 q^{18} + 4 q^{19} + 16 q^{21} + 3 q^{22} + 6 q^{23} - 6 q^{26} - q^{28} + 11 q^{29} + 23 q^{31} + 5 q^{32} + 20 q^{33} + 9 q^{34} + 3 q^{36} + 5 q^{37} + 4 q^{38} + 20 q^{39} - 7 q^{41} + 16 q^{42} - 17 q^{43} + 3 q^{44} + 6 q^{46} + 12 q^{47} + 30 q^{49} - 20 q^{51} - 6 q^{52} - 7 q^{53} - q^{56} + 11 q^{58} + 20 q^{59} - 9 q^{61} + 23 q^{62} - 33 q^{63} + 5 q^{64} + 20 q^{66} + 12 q^{67} + 9 q^{68} + 16 q^{69} + 6 q^{71} + 3 q^{72} - 6 q^{73} + 5 q^{74} + 4 q^{76} - q^{77} + 20 q^{78} + 20 q^{79} - 7 q^{81} - 7 q^{82} + 12 q^{83} + 16 q^{84} - 17 q^{86} - 34 q^{87} + 3 q^{88} + 12 q^{89} + 16 q^{91} + 6 q^{92} - 4 q^{93} + 12 q^{94} + 3 q^{97} + 30 q^{98} - 11 q^{99}+O(q^{100})$$ 5 * q + 5 * q^2 + 5 * q^4 - q^7 + 5 * q^8 + 3 * q^9 + 3 * q^11 - 6 * q^13 - q^14 + 5 * q^16 + 9 * q^17 + 3 * q^18 + 4 * q^19 + 16 * q^21 + 3 * q^22 + 6 * q^23 - 6 * q^26 - q^28 + 11 * q^29 + 23 * q^31 + 5 * q^32 + 20 * q^33 + 9 * q^34 + 3 * q^36 + 5 * q^37 + 4 * q^38 + 20 * q^39 - 7 * q^41 + 16 * q^42 - 17 * q^43 + 3 * q^44 + 6 * q^46 + 12 * q^47 + 30 * q^49 - 20 * q^51 - 6 * q^52 - 7 * q^53 - q^56 + 11 * q^58 + 20 * q^59 - 9 * q^61 + 23 * q^62 - 33 * q^63 + 5 * q^64 + 20 * q^66 + 12 * q^67 + 9 * q^68 + 16 * q^69 + 6 * q^71 + 3 * q^72 - 6 * q^73 + 5 * q^74 + 4 * q^76 - q^77 + 20 * q^78 + 20 * q^79 - 7 * q^81 - 7 * q^82 + 12 * q^83 + 16 * q^84 - 17 * q^86 - 34 * q^87 + 3 * q^88 + 12 * q^89 + 16 * q^91 + 6 * q^92 - 4 * q^93 + 12 * q^94 + 3 * q^97 + 30 * q^98 - 11 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.53175 0.884356 0.442178 0.896927i $$-0.354206\pi$$
0.442178 + 0.896927i $$0.354206\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.53175 0.625334
$$7$$ 4.67211 1.76589 0.882946 0.469475i $$-0.155557\pi$$
0.882946 + 0.469475i $$0.155557\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −0.653743 −0.217914
$$10$$ 0 0
$$11$$ 0.0451320 0.0136078 0.00680390 0.999977i $$-0.497834\pi$$
0.00680390 + 0.999977i $$0.497834\pi$$
$$12$$ 1.53175 0.442178
$$13$$ 5.26514 1.46029 0.730143 0.683294i $$-0.239453\pi$$
0.730143 + 0.683294i $$0.239453\pi$$
$$14$$ 4.67211 1.24867
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.60861 0.875217 0.437608 0.899166i $$-0.355826\pi$$
0.437608 + 0.899166i $$0.355826\pi$$
$$18$$ −0.653743 −0.154089
$$19$$ −6.22000 −1.42697 −0.713483 0.700672i $$-0.752884\pi$$
−0.713483 + 0.700672i $$0.752884\pi$$
$$20$$ 0 0
$$21$$ 7.15650 1.56168
$$22$$ 0.0451320 0.00962217
$$23$$ −2.20164 −0.459073 −0.229536 0.973300i $$-0.573721\pi$$
−0.229536 + 0.973300i $$0.573721\pi$$
$$24$$ 1.53175 0.312667
$$25$$ 0 0
$$26$$ 5.26514 1.03258
$$27$$ −5.59662 −1.07707
$$28$$ 4.67211 0.882946
$$29$$ −4.20386 −0.780637 −0.390319 0.920680i $$-0.627635\pi$$
−0.390319 + 0.920680i $$0.627635\pi$$
$$30$$ 0 0
$$31$$ −3.01051 −0.540704 −0.270352 0.962762i $$-0.587140\pi$$
−0.270352 + 0.962762i $$0.587140\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0.0691309 0.0120341
$$34$$ 3.60861 0.618872
$$35$$ 0 0
$$36$$ −0.653743 −0.108957
$$37$$ 1.00000 0.164399
$$38$$ −6.22000 −1.00902
$$39$$ 8.06487 1.29141
$$40$$ 0 0
$$41$$ −7.38299 −1.15303 −0.576515 0.817087i $$-0.695588\pi$$
−0.576515 + 0.817087i $$0.695588\pi$$
$$42$$ 7.15650 1.10427
$$43$$ −5.54789 −0.846046 −0.423023 0.906119i $$-0.639031\pi$$
−0.423023 + 0.906119i $$0.639031\pi$$
$$44$$ 0.0451320 0.00680390
$$45$$ 0 0
$$46$$ −2.20164 −0.324613
$$47$$ −4.28072 −0.624407 −0.312204 0.950015i $$-0.601067\pi$$
−0.312204 + 0.950015i $$0.601067\pi$$
$$48$$ 1.53175 0.221089
$$49$$ 14.8286 2.11837
$$50$$ 0 0
$$51$$ 5.52749 0.774003
$$52$$ 5.26514 0.730143
$$53$$ 6.10215 0.838194 0.419097 0.907941i $$-0.362347\pi$$
0.419097 + 0.907941i $$0.362347\pi$$
$$54$$ −5.59662 −0.761603
$$55$$ 0 0
$$56$$ 4.67211 0.624337
$$57$$ −9.52749 −1.26195
$$58$$ −4.20386 −0.551994
$$59$$ 13.5275 1.76113 0.880565 0.473926i $$-0.157164\pi$$
0.880565 + 0.473926i $$0.157164\pi$$
$$60$$ 0 0
$$61$$ −4.27299 −0.547100 −0.273550 0.961858i $$-0.588198\pi$$
−0.273550 + 0.961858i $$0.588198\pi$$
$$62$$ −3.01051 −0.382335
$$63$$ −3.05436 −0.384813
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0.0691309 0.00850942
$$67$$ 12.3751 1.51186 0.755932 0.654650i $$-0.227184\pi$$
0.755932 + 0.654650i $$0.227184\pi$$
$$68$$ 3.60861 0.437608
$$69$$ −3.37235 −0.405984
$$70$$ 0 0
$$71$$ −4.49354 −0.533285 −0.266642 0.963796i $$-0.585914\pi$$
−0.266642 + 0.963796i $$0.585914\pi$$
$$72$$ −0.653743 −0.0770443
$$73$$ 7.03811 0.823748 0.411874 0.911241i $$-0.364874\pi$$
0.411874 + 0.911241i $$0.364874\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −6.22000 −0.713483
$$77$$ 0.210861 0.0240299
$$78$$ 8.06487 0.913167
$$79$$ −8.72499 −0.981638 −0.490819 0.871262i $$-0.663302\pi$$
−0.490819 + 0.871262i $$0.663302\pi$$
$$80$$ 0 0
$$81$$ −6.61139 −0.734599
$$82$$ −7.38299 −0.815315
$$83$$ −0.880231 −0.0966179 −0.0483090 0.998832i $$-0.515383\pi$$
−0.0483090 + 0.998832i $$0.515383\pi$$
$$84$$ 7.15650 0.780839
$$85$$ 0 0
$$86$$ −5.54789 −0.598245
$$87$$ −6.43926 −0.690361
$$88$$ 0.0451320 0.00481108
$$89$$ 9.97602 1.05746 0.528728 0.848791i $$-0.322669\pi$$
0.528728 + 0.848791i $$0.322669\pi$$
$$90$$ 0 0
$$91$$ 24.5993 2.57871
$$92$$ −2.20164 −0.229536
$$93$$ −4.61135 −0.478175
$$94$$ −4.28072 −0.441523
$$95$$ 0 0
$$96$$ 1.53175 0.156334
$$97$$ 0.240408 0.0244097 0.0122049 0.999926i $$-0.496115\pi$$
0.0122049 + 0.999926i $$0.496115\pi$$
$$98$$ 14.8286 1.49792
$$99$$ −0.0295047 −0.00296533
$$100$$ 0 0
$$101$$ −5.61775 −0.558987 −0.279494 0.960148i $$-0.590167\pi$$
−0.279494 + 0.960148i $$0.590167\pi$$
$$102$$ 5.52749 0.547303
$$103$$ −12.5663 −1.23819 −0.619095 0.785316i $$-0.712501\pi$$
−0.619095 + 0.785316i $$0.712501\pi$$
$$104$$ 5.26514 0.516289
$$105$$ 0 0
$$106$$ 6.10215 0.592693
$$107$$ −17.9395 −1.73427 −0.867137 0.498070i $$-0.834042\pi$$
−0.867137 + 0.498070i $$0.834042\pi$$
$$108$$ −5.59662 −0.538535
$$109$$ 3.09936 0.296865 0.148433 0.988923i $$-0.452577\pi$$
0.148433 + 0.988923i $$0.452577\pi$$
$$110$$ 0 0
$$111$$ 1.53175 0.145387
$$112$$ 4.67211 0.441473
$$113$$ 4.36184 0.410328 0.205164 0.978728i $$-0.434227\pi$$
0.205164 + 0.978728i $$0.434227\pi$$
$$114$$ −9.52749 −0.892331
$$115$$ 0 0
$$116$$ −4.20386 −0.390319
$$117$$ −3.44204 −0.318217
$$118$$ 13.5275 1.24531
$$119$$ 16.8598 1.54554
$$120$$ 0 0
$$121$$ −10.9980 −0.999815
$$122$$ −4.27299 −0.386858
$$123$$ −11.3089 −1.01969
$$124$$ −3.01051 −0.270352
$$125$$ 0 0
$$126$$ −3.05436 −0.272104
$$127$$ 17.6572 1.56683 0.783413 0.621502i $$-0.213477\pi$$
0.783413 + 0.621502i $$0.213477\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −8.49798 −0.748206
$$130$$ 0 0
$$131$$ −10.3103 −0.900812 −0.450406 0.892824i $$-0.648721\pi$$
−0.450406 + 0.892824i $$0.648721\pi$$
$$132$$ 0.0691309 0.00601707
$$133$$ −29.0605 −2.51987
$$134$$ 12.3751 1.06905
$$135$$ 0 0
$$136$$ 3.60861 0.309436
$$137$$ 3.21863 0.274986 0.137493 0.990503i $$-0.456095\pi$$
0.137493 + 0.990503i $$0.456095\pi$$
$$138$$ −3.37235 −0.287074
$$139$$ 20.9400 1.77611 0.888055 0.459737i $$-0.152056\pi$$
0.888055 + 0.459737i $$0.152056\pi$$
$$140$$ 0 0
$$141$$ −6.55699 −0.552198
$$142$$ −4.49354 −0.377089
$$143$$ 0.237626 0.0198713
$$144$$ −0.653743 −0.0544786
$$145$$ 0 0
$$146$$ 7.03811 0.582478
$$147$$ 22.7137 1.87340
$$148$$ 1.00000 0.0821995
$$149$$ −21.9832 −1.80093 −0.900467 0.434924i $$-0.856775\pi$$
−0.900467 + 0.434924i $$0.856775\pi$$
$$150$$ 0 0
$$151$$ 14.1877 1.15458 0.577290 0.816539i $$-0.304110\pi$$
0.577290 + 0.816539i $$0.304110\pi$$
$$152$$ −6.22000 −0.504509
$$153$$ −2.35910 −0.190722
$$154$$ 0.210861 0.0169917
$$155$$ 0 0
$$156$$ 8.06487 0.645706
$$157$$ −19.9316 −1.59072 −0.795359 0.606139i $$-0.792717\pi$$
−0.795359 + 0.606139i $$0.792717\pi$$
$$158$$ −8.72499 −0.694123
$$159$$ 9.34696 0.741262
$$160$$ 0 0
$$161$$ −10.2863 −0.810673
$$162$$ −6.61139 −0.519440
$$163$$ 8.65087 0.677588 0.338794 0.940861i $$-0.389981\pi$$
0.338794 + 0.940861i $$0.389981\pi$$
$$164$$ −7.38299 −0.576515
$$165$$ 0 0
$$166$$ −0.880231 −0.0683192
$$167$$ −11.9640 −0.925803 −0.462901 0.886410i $$-0.653192\pi$$
−0.462901 + 0.886410i $$0.653192\pi$$
$$168$$ 7.15650 0.552136
$$169$$ 14.7216 1.13243
$$170$$ 0 0
$$171$$ 4.06628 0.310956
$$172$$ −5.54789 −0.423023
$$173$$ −5.33508 −0.405619 −0.202809 0.979218i $$-0.565007\pi$$
−0.202809 + 0.979218i $$0.565007\pi$$
$$174$$ −6.43926 −0.488159
$$175$$ 0 0
$$176$$ 0.0451320 0.00340195
$$177$$ 20.7207 1.55747
$$178$$ 9.97602 0.747734
$$179$$ −5.27905 −0.394575 −0.197288 0.980346i $$-0.563213\pi$$
−0.197288 + 0.980346i $$0.563213\pi$$
$$180$$ 0 0
$$181$$ 17.7448 1.31896 0.659478 0.751723i $$-0.270777\pi$$
0.659478 + 0.751723i $$0.270777\pi$$
$$182$$ 24.5993 1.82342
$$183$$ −6.54515 −0.483832
$$184$$ −2.20164 −0.162307
$$185$$ 0 0
$$186$$ −4.61135 −0.338121
$$187$$ 0.162864 0.0119098
$$188$$ −4.28072 −0.312204
$$189$$ −26.1480 −1.90199
$$190$$ 0 0
$$191$$ −3.11649 −0.225501 −0.112751 0.993623i $$-0.535966\pi$$
−0.112751 + 0.993623i $$0.535966\pi$$
$$192$$ 1.53175 0.110545
$$193$$ −8.78278 −0.632198 −0.316099 0.948726i $$-0.602373\pi$$
−0.316099 + 0.948726i $$0.602373\pi$$
$$194$$ 0.240408 0.0172603
$$195$$ 0 0
$$196$$ 14.8286 1.05919
$$197$$ 7.00278 0.498928 0.249464 0.968384i $$-0.419746\pi$$
0.249464 + 0.968384i $$0.419746\pi$$
$$198$$ −0.0295047 −0.00209681
$$199$$ 17.3202 1.22780 0.613900 0.789384i $$-0.289600\pi$$
0.613900 + 0.789384i $$0.289600\pi$$
$$200$$ 0 0
$$201$$ 18.9556 1.33703
$$202$$ −5.61775 −0.395264
$$203$$ −19.6409 −1.37852
$$204$$ 5.52749 0.387002
$$205$$ 0 0
$$206$$ −12.5663 −0.875533
$$207$$ 1.43930 0.100038
$$208$$ 5.26514 0.365071
$$209$$ −0.280721 −0.0194179
$$210$$ 0 0
$$211$$ 6.77438 0.466368 0.233184 0.972433i $$-0.425086\pi$$
0.233184 + 0.972433i $$0.425086\pi$$
$$212$$ 6.10215 0.419097
$$213$$ −6.88297 −0.471613
$$214$$ −17.9395 −1.22632
$$215$$ 0 0
$$216$$ −5.59662 −0.380802
$$217$$ −14.0654 −0.954825
$$218$$ 3.09936 0.209915
$$219$$ 10.7806 0.728487
$$220$$ 0 0
$$221$$ 18.9998 1.27807
$$222$$ 1.53175 0.102804
$$223$$ −10.2559 −0.686784 −0.343392 0.939192i $$-0.611576\pi$$
−0.343392 + 0.939192i $$0.611576\pi$$
$$224$$ 4.67211 0.312168
$$225$$ 0 0
$$226$$ 4.36184 0.290145
$$227$$ −11.2744 −0.748306 −0.374153 0.927367i $$-0.622066\pi$$
−0.374153 + 0.927367i $$0.622066\pi$$
$$228$$ −9.52749 −0.630973
$$229$$ −9.40494 −0.621496 −0.310748 0.950492i $$-0.600579\pi$$
−0.310748 + 0.950492i $$0.600579\pi$$
$$230$$ 0 0
$$231$$ 0.322987 0.0212510
$$232$$ −4.20386 −0.275997
$$233$$ −3.22867 −0.211517 −0.105759 0.994392i $$-0.533727\pi$$
−0.105759 + 0.994392i $$0.533727\pi$$
$$234$$ −3.44204 −0.225013
$$235$$ 0 0
$$236$$ 13.5275 0.880565
$$237$$ −13.3645 −0.868118
$$238$$ 16.8598 1.09286
$$239$$ −17.8612 −1.15534 −0.577672 0.816269i $$-0.696039\pi$$
−0.577672 + 0.816269i $$0.696039\pi$$
$$240$$ 0 0
$$241$$ 26.6840 1.71887 0.859434 0.511248i $$-0.170816\pi$$
0.859434 + 0.511248i $$0.170816\pi$$
$$242$$ −10.9980 −0.706976
$$243$$ 6.66286 0.427423
$$244$$ −4.27299 −0.273550
$$245$$ 0 0
$$246$$ −11.3089 −0.721029
$$247$$ −32.7492 −2.08378
$$248$$ −3.01051 −0.191168
$$249$$ −1.34829 −0.0854447
$$250$$ 0 0
$$251$$ −30.1552 −1.90338 −0.951690 0.307060i $$-0.900655\pi$$
−0.951690 + 0.307060i $$0.900655\pi$$
$$252$$ −3.05436 −0.192406
$$253$$ −0.0993641 −0.00624697
$$254$$ 17.6572 1.10791
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −3.15376 −0.196726 −0.0983632 0.995151i $$-0.531361\pi$$
−0.0983632 + 0.995151i $$0.531361\pi$$
$$258$$ −8.49798 −0.529061
$$259$$ 4.67211 0.290311
$$260$$ 0 0
$$261$$ 2.74824 0.170112
$$262$$ −10.3103 −0.636970
$$263$$ 14.0332 0.865322 0.432661 0.901557i $$-0.357575\pi$$
0.432661 + 0.901557i $$0.357575\pi$$
$$264$$ 0.0691309 0.00425471
$$265$$ 0 0
$$266$$ −29.0605 −1.78182
$$267$$ 15.2808 0.935167
$$268$$ 12.3751 0.755932
$$269$$ −30.1257 −1.83679 −0.918397 0.395660i $$-0.870516\pi$$
−0.918397 + 0.395660i $$0.870516\pi$$
$$270$$ 0 0
$$271$$ 14.8802 0.903910 0.451955 0.892041i $$-0.350727\pi$$
0.451955 + 0.892041i $$0.350727\pi$$
$$272$$ 3.60861 0.218804
$$273$$ 37.6800 2.28049
$$274$$ 3.21863 0.194445
$$275$$ 0 0
$$276$$ −3.37235 −0.202992
$$277$$ 0.147278 0.00884908 0.00442454 0.999990i $$-0.498592\pi$$
0.00442454 + 0.999990i $$0.498592\pi$$
$$278$$ 20.9400 1.25590
$$279$$ 1.96810 0.117827
$$280$$ 0 0
$$281$$ −25.5247 −1.52268 −0.761339 0.648354i $$-0.775458\pi$$
−0.761339 + 0.648354i $$0.775458\pi$$
$$282$$ −6.55699 −0.390463
$$283$$ 0.0562691 0.00334485 0.00167242 0.999999i $$-0.499468\pi$$
0.00167242 + 0.999999i $$0.499468\pi$$
$$284$$ −4.49354 −0.266642
$$285$$ 0 0
$$286$$ 0.237626 0.0140511
$$287$$ −34.4942 −2.03613
$$288$$ −0.653743 −0.0385222
$$289$$ −3.97793 −0.233996
$$290$$ 0 0
$$291$$ 0.368245 0.0215869
$$292$$ 7.03811 0.411874
$$293$$ 13.7033 0.800557 0.400278 0.916394i $$-0.368913\pi$$
0.400278 + 0.916394i $$0.368913\pi$$
$$294$$ 22.7137 1.32469
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ −0.252586 −0.0146565
$$298$$ −21.9832 −1.27345
$$299$$ −11.5919 −0.670377
$$300$$ 0 0
$$301$$ −25.9204 −1.49403
$$302$$ 14.1877 0.816411
$$303$$ −8.60499 −0.494344
$$304$$ −6.22000 −0.356742
$$305$$ 0 0
$$306$$ −2.35910 −0.134861
$$307$$ −17.5823 −1.00348 −0.501738 0.865020i $$-0.667306\pi$$
−0.501738 + 0.865020i $$0.667306\pi$$
$$308$$ 0.210861 0.0120149
$$309$$ −19.2484 −1.09500
$$310$$ 0 0
$$311$$ 0.353133 0.0200244 0.0100122 0.999950i $$-0.496813\pi$$
0.0100122 + 0.999950i $$0.496813\pi$$
$$312$$ 8.06487 0.456583
$$313$$ −18.8717 −1.06669 −0.533346 0.845897i $$-0.679066\pi$$
−0.533346 + 0.845897i $$0.679066\pi$$
$$314$$ −19.9316 −1.12481
$$315$$ 0 0
$$316$$ −8.72499 −0.490819
$$317$$ 15.1979 0.853601 0.426800 0.904346i $$-0.359641\pi$$
0.426800 + 0.904346i $$0.359641\pi$$
$$318$$ 9.34696 0.524152
$$319$$ −0.189728 −0.0106228
$$320$$ 0 0
$$321$$ −27.4788 −1.53372
$$322$$ −10.2863 −0.573232
$$323$$ −22.4456 −1.24890
$$324$$ −6.61139 −0.367300
$$325$$ 0 0
$$326$$ 8.65087 0.479127
$$327$$ 4.74745 0.262535
$$328$$ −7.38299 −0.407658
$$329$$ −20.0000 −1.10264
$$330$$ 0 0
$$331$$ −23.0210 −1.26535 −0.632674 0.774418i $$-0.718043\pi$$
−0.632674 + 0.774418i $$0.718043\pi$$
$$332$$ −0.880231 −0.0483090
$$333$$ −0.653743 −0.0358249
$$334$$ −11.9640 −0.654641
$$335$$ 0 0
$$336$$ 7.15650 0.390419
$$337$$ 34.0776 1.85633 0.928163 0.372173i $$-0.121387\pi$$
0.928163 + 0.372173i $$0.121387\pi$$
$$338$$ 14.7216 0.800752
$$339$$ 6.68125 0.362876
$$340$$ 0 0
$$341$$ −0.135870 −0.00735779
$$342$$ 4.06628 0.219879
$$343$$ 36.5761 1.97493
$$344$$ −5.54789 −0.299122
$$345$$ 0 0
$$346$$ −5.33508 −0.286816
$$347$$ 13.8449 0.743236 0.371618 0.928386i $$-0.378803\pi$$
0.371618 + 0.928386i $$0.378803\pi$$
$$348$$ −6.43926 −0.345181
$$349$$ 11.3765 0.608970 0.304485 0.952517i $$-0.401516\pi$$
0.304485 + 0.952517i $$0.401516\pi$$
$$350$$ 0 0
$$351$$ −29.4670 −1.57283
$$352$$ 0.0451320 0.00240554
$$353$$ 19.2631 1.02527 0.512636 0.858606i $$-0.328669\pi$$
0.512636 + 0.858606i $$0.328669\pi$$
$$354$$ 20.7207 1.10129
$$355$$ 0 0
$$356$$ 9.97602 0.528728
$$357$$ 25.8250 1.36681
$$358$$ −5.27905 −0.279007
$$359$$ 14.6645 0.773961 0.386980 0.922088i $$-0.373518\pi$$
0.386980 + 0.922088i $$0.373518\pi$$
$$360$$ 0 0
$$361$$ 19.6884 1.03623
$$362$$ 17.7448 0.932643
$$363$$ −16.8461 −0.884192
$$364$$ 24.5993 1.28935
$$365$$ 0 0
$$366$$ −6.54515 −0.342121
$$367$$ 0.868349 0.0453275 0.0226637 0.999743i $$-0.492785\pi$$
0.0226637 + 0.999743i $$0.492785\pi$$
$$368$$ −2.20164 −0.114768
$$369$$ 4.82658 0.251262
$$370$$ 0 0
$$371$$ 28.5099 1.48016
$$372$$ −4.61135 −0.239087
$$373$$ 32.4555 1.68048 0.840240 0.542214i $$-0.182414\pi$$
0.840240 + 0.542214i $$0.182414\pi$$
$$374$$ 0.162864 0.00842148
$$375$$ 0 0
$$376$$ −4.28072 −0.220761
$$377$$ −22.1339 −1.13995
$$378$$ −26.1480 −1.34491
$$379$$ −30.2511 −1.55389 −0.776947 0.629566i $$-0.783233\pi$$
−0.776947 + 0.629566i $$0.783233\pi$$
$$380$$ 0 0
$$381$$ 27.0465 1.38563
$$382$$ −3.11649 −0.159453
$$383$$ 14.4935 0.740585 0.370293 0.928915i $$-0.379257\pi$$
0.370293 + 0.928915i $$0.379257\pi$$
$$384$$ 1.53175 0.0781668
$$385$$ 0 0
$$386$$ −8.78278 −0.447032
$$387$$ 3.62689 0.184365
$$388$$ 0.240408 0.0122049
$$389$$ 36.9033 1.87107 0.935537 0.353229i $$-0.114916\pi$$
0.935537 + 0.353229i $$0.114916\pi$$
$$390$$ 0 0
$$391$$ −7.94485 −0.401788
$$392$$ 14.8286 0.748958
$$393$$ −15.7927 −0.796639
$$394$$ 7.00278 0.352795
$$395$$ 0 0
$$396$$ −0.0295047 −0.00148267
$$397$$ 22.9365 1.15115 0.575575 0.817749i $$-0.304778\pi$$
0.575575 + 0.817749i $$0.304778\pi$$
$$398$$ 17.3202 0.868185
$$399$$ −44.5135 −2.22846
$$400$$ 0 0
$$401$$ 27.4332 1.36995 0.684973 0.728568i $$-0.259814\pi$$
0.684973 + 0.728568i $$0.259814\pi$$
$$402$$ 18.9556 0.945420
$$403$$ −15.8508 −0.789582
$$404$$ −5.61775 −0.279494
$$405$$ 0 0
$$406$$ −19.6409 −0.974761
$$407$$ 0.0451320 0.00223711
$$408$$ 5.52749 0.273651
$$409$$ −21.3920 −1.05777 −0.528883 0.848695i $$-0.677389\pi$$
−0.528883 + 0.848695i $$0.677389\pi$$
$$410$$ 0 0
$$411$$ 4.93014 0.243186
$$412$$ −12.5663 −0.619095
$$413$$ 63.2019 3.10996
$$414$$ 1.43930 0.0707379
$$415$$ 0 0
$$416$$ 5.26514 0.258144
$$417$$ 32.0749 1.57071
$$418$$ −0.280721 −0.0137305
$$419$$ 23.0891 1.12797 0.563987 0.825784i $$-0.309267\pi$$
0.563987 + 0.825784i $$0.309267\pi$$
$$420$$ 0 0
$$421$$ −1.83757 −0.0895576 −0.0447788 0.998997i $$-0.514258\pi$$
−0.0447788 + 0.998997i $$0.514258\pi$$
$$422$$ 6.77438 0.329772
$$423$$ 2.79849 0.136067
$$424$$ 6.10215 0.296346
$$425$$ 0 0
$$426$$ −6.88297 −0.333481
$$427$$ −19.9639 −0.966120
$$428$$ −17.9395 −0.867137
$$429$$ 0.363983 0.0175733
$$430$$ 0 0
$$431$$ −15.9796 −0.769710 −0.384855 0.922977i $$-0.625749\pi$$
−0.384855 + 0.922977i $$0.625749\pi$$
$$432$$ −5.59662 −0.269267
$$433$$ −25.2525 −1.21356 −0.606779 0.794870i $$-0.707539\pi$$
−0.606779 + 0.794870i $$0.707539\pi$$
$$434$$ −14.0654 −0.675163
$$435$$ 0 0
$$436$$ 3.09936 0.148433
$$437$$ 13.6942 0.655082
$$438$$ 10.7806 0.515118
$$439$$ −9.30792 −0.444243 −0.222121 0.975019i $$-0.571298\pi$$
−0.222121 + 0.975019i $$0.571298\pi$$
$$440$$ 0 0
$$441$$ −9.69410 −0.461624
$$442$$ 18.9998 0.903729
$$443$$ −9.52445 −0.452520 −0.226260 0.974067i $$-0.572650\pi$$
−0.226260 + 0.974067i $$0.572650\pi$$
$$444$$ 1.53175 0.0726936
$$445$$ 0 0
$$446$$ −10.2559 −0.485629
$$447$$ −33.6728 −1.59267
$$448$$ 4.67211 0.220736
$$449$$ 5.43278 0.256389 0.128194 0.991749i $$-0.459082\pi$$
0.128194 + 0.991749i $$0.459082\pi$$
$$450$$ 0 0
$$451$$ −0.333209 −0.0156902
$$452$$ 4.36184 0.205164
$$453$$ 21.7320 1.02106
$$454$$ −11.2744 −0.529132
$$455$$ 0 0
$$456$$ −9.52749 −0.446166
$$457$$ 7.54789 0.353076 0.176538 0.984294i $$-0.443510\pi$$
0.176538 + 0.984294i $$0.443510\pi$$
$$458$$ −9.40494 −0.439464
$$459$$ −20.1960 −0.942670
$$460$$ 0 0
$$461$$ −35.4323 −1.65025 −0.825124 0.564952i $$-0.808895\pi$$
−0.825124 + 0.564952i $$0.808895\pi$$
$$462$$ 0.322987 0.0150267
$$463$$ 20.9124 0.971883 0.485942 0.873991i $$-0.338477\pi$$
0.485942 + 0.873991i $$0.338477\pi$$
$$464$$ −4.20386 −0.195159
$$465$$ 0 0
$$466$$ −3.22867 −0.149565
$$467$$ 6.62523 0.306579 0.153289 0.988181i $$-0.451013\pi$$
0.153289 + 0.988181i $$0.451013\pi$$
$$468$$ −3.44204 −0.159109
$$469$$ 57.8180 2.66979
$$470$$ 0 0
$$471$$ −30.5303 −1.40676
$$472$$ 13.5275 0.622653
$$473$$ −0.250387 −0.0115128
$$474$$ −13.3645 −0.613852
$$475$$ 0 0
$$476$$ 16.8598 0.772769
$$477$$ −3.98923 −0.182654
$$478$$ −17.8612 −0.816952
$$479$$ −29.4581 −1.34597 −0.672987 0.739655i $$-0.734989\pi$$
−0.672987 + 0.739655i $$0.734989\pi$$
$$480$$ 0 0
$$481$$ 5.26514 0.240070
$$482$$ 26.6840 1.21542
$$483$$ −15.7560 −0.716923
$$484$$ −10.9980 −0.499907
$$485$$ 0 0
$$486$$ 6.66286 0.302233
$$487$$ 8.80654 0.399063 0.199531 0.979891i $$-0.436058\pi$$
0.199531 + 0.979891i $$0.436058\pi$$
$$488$$ −4.27299 −0.193429
$$489$$ 13.2510 0.599229
$$490$$ 0 0
$$491$$ 5.14258 0.232082 0.116041 0.993244i $$-0.462980\pi$$
0.116041 + 0.993244i $$0.462980\pi$$
$$492$$ −11.3089 −0.509844
$$493$$ −15.1701 −0.683227
$$494$$ −32.7492 −1.47345
$$495$$ 0 0
$$496$$ −3.01051 −0.135176
$$497$$ −20.9943 −0.941723
$$498$$ −1.34829 −0.0604185
$$499$$ −12.9692 −0.580579 −0.290290 0.956939i $$-0.593752\pi$$
−0.290290 + 0.956939i $$0.593752\pi$$
$$500$$ 0 0
$$501$$ −18.3259 −0.818739
$$502$$ −30.1552 −1.34589
$$503$$ 17.2091 0.767314 0.383657 0.923476i $$-0.374664\pi$$
0.383657 + 0.923476i $$0.374664\pi$$
$$504$$ −3.05436 −0.136052
$$505$$ 0 0
$$506$$ −0.0993641 −0.00441727
$$507$$ 22.5499 1.00148
$$508$$ 17.6572 0.783413
$$509$$ −12.1085 −0.536698 −0.268349 0.963322i $$-0.586478\pi$$
−0.268349 + 0.963322i $$0.586478\pi$$
$$510$$ 0 0
$$511$$ 32.8828 1.45465
$$512$$ 1.00000 0.0441942
$$513$$ 34.8110 1.53694
$$514$$ −3.15376 −0.139107
$$515$$ 0 0
$$516$$ −8.49798 −0.374103
$$517$$ −0.193197 −0.00849681
$$518$$ 4.67211 0.205281
$$519$$ −8.17201 −0.358711
$$520$$ 0 0
$$521$$ −18.7936 −0.823362 −0.411681 0.911328i $$-0.635058\pi$$
−0.411681 + 0.911328i $$0.635058\pi$$
$$522$$ 2.74824 0.120287
$$523$$ −42.9985 −1.88019 −0.940096 0.340909i $$-0.889265\pi$$
−0.940096 + 0.340909i $$0.889265\pi$$
$$524$$ −10.3103 −0.450406
$$525$$ 0 0
$$526$$ 14.0332 0.611875
$$527$$ −10.8638 −0.473233
$$528$$ 0.0691309 0.00300853
$$529$$ −18.1528 −0.789252
$$530$$ 0 0
$$531$$ −8.84350 −0.383775
$$532$$ −29.0605 −1.25993
$$533$$ −38.8725 −1.68375
$$534$$ 15.2808 0.661263
$$535$$ 0 0
$$536$$ 12.3751 0.534525
$$537$$ −8.08619 −0.348945
$$538$$ −30.1257 −1.29881
$$539$$ 0.669244 0.0288264
$$540$$ 0 0
$$541$$ 30.1751 1.29733 0.648664 0.761075i $$-0.275328\pi$$
0.648664 + 0.761075i $$0.275328\pi$$
$$542$$ 14.8802 0.639161
$$543$$ 27.1805 1.16643
$$544$$ 3.60861 0.154718
$$545$$ 0 0
$$546$$ 37.6800 1.61255
$$547$$ −31.6736 −1.35426 −0.677132 0.735862i $$-0.736777\pi$$
−0.677132 + 0.735862i $$0.736777\pi$$
$$548$$ 3.21863 0.137493
$$549$$ 2.79344 0.119221
$$550$$ 0 0
$$551$$ 26.1480 1.11394
$$552$$ −3.37235 −0.143537
$$553$$ −40.7641 −1.73347
$$554$$ 0.147278 0.00625724
$$555$$ 0 0
$$556$$ 20.9400 0.888055
$$557$$ −0.708971 −0.0300401 −0.0150200 0.999887i $$-0.504781\pi$$
−0.0150200 + 0.999887i $$0.504781\pi$$
$$558$$ 1.96810 0.0833163
$$559$$ −29.2104 −1.23547
$$560$$ 0 0
$$561$$ 0.249466 0.0105325
$$562$$ −25.5247 −1.07670
$$563$$ −29.9399 −1.26182 −0.630908 0.775857i $$-0.717318\pi$$
−0.630908 + 0.775857i $$0.717318\pi$$
$$564$$ −6.55699 −0.276099
$$565$$ 0 0
$$566$$ 0.0562691 0.00236517
$$567$$ −30.8892 −1.29722
$$568$$ −4.49354 −0.188545
$$569$$ 21.1453 0.886456 0.443228 0.896409i $$-0.353833\pi$$
0.443228 + 0.896409i $$0.353833\pi$$
$$570$$ 0 0
$$571$$ −19.5326 −0.817416 −0.408708 0.912665i $$-0.634020\pi$$
−0.408708 + 0.912665i $$0.634020\pi$$
$$572$$ 0.237626 0.00993564
$$573$$ −4.77368 −0.199423
$$574$$ −34.4942 −1.43976
$$575$$ 0 0
$$576$$ −0.653743 −0.0272393
$$577$$ 1.08303 0.0450873 0.0225436 0.999746i $$-0.492824\pi$$
0.0225436 + 0.999746i $$0.492824\pi$$
$$578$$ −3.97793 −0.165460
$$579$$ −13.4530 −0.559088
$$580$$ 0 0
$$581$$ −4.11254 −0.170617
$$582$$ 0.368245 0.0152642
$$583$$ 0.275402 0.0114060
$$584$$ 7.03811 0.291239
$$585$$ 0 0
$$586$$ 13.7033 0.566079
$$587$$ 1.16839 0.0482244 0.0241122 0.999709i $$-0.492324\pi$$
0.0241122 + 0.999709i $$0.492324\pi$$
$$588$$ 22.7137 0.936698
$$589$$ 18.7254 0.771566
$$590$$ 0 0
$$591$$ 10.7265 0.441230
$$592$$ 1.00000 0.0410997
$$593$$ −6.01305 −0.246926 −0.123463 0.992349i $$-0.539400\pi$$
−0.123463 + 0.992349i $$0.539400\pi$$
$$594$$ −0.252586 −0.0103637
$$595$$ 0 0
$$596$$ −21.9832 −0.900467
$$597$$ 26.5303 1.08581
$$598$$ −11.5919 −0.474028
$$599$$ −11.9208 −0.487072 −0.243536 0.969892i $$-0.578307\pi$$
−0.243536 + 0.969892i $$0.578307\pi$$
$$600$$ 0 0
$$601$$ 21.7772 0.888309 0.444155 0.895950i $$-0.353504\pi$$
0.444155 + 0.895950i $$0.353504\pi$$
$$602$$ −25.9204 −1.05644
$$603$$ −8.09015 −0.329457
$$604$$ 14.1877 0.577290
$$605$$ 0 0
$$606$$ −8.60499 −0.349554
$$607$$ 5.16486 0.209635 0.104818 0.994491i $$-0.466574\pi$$
0.104818 + 0.994491i $$0.466574\pi$$
$$608$$ −6.22000 −0.252254
$$609$$ −30.0849 −1.21910
$$610$$ 0 0
$$611$$ −22.5386 −0.911813
$$612$$ −2.35910 −0.0953611
$$613$$ −39.5817 −1.59869 −0.799344 0.600873i $$-0.794820\pi$$
−0.799344 + 0.600873i $$0.794820\pi$$
$$614$$ −17.5823 −0.709565
$$615$$ 0 0
$$616$$ 0.210861 0.00849585
$$617$$ 44.9917 1.81130 0.905650 0.424027i $$-0.139384\pi$$
0.905650 + 0.424027i $$0.139384\pi$$
$$618$$ −19.2484 −0.774283
$$619$$ 30.0465 1.20767 0.603836 0.797108i $$-0.293638\pi$$
0.603836 + 0.797108i $$0.293638\pi$$
$$620$$ 0 0
$$621$$ 12.3217 0.494453
$$622$$ 0.353133 0.0141594
$$623$$ 46.6091 1.86735
$$624$$ 8.06487 0.322853
$$625$$ 0 0
$$626$$ −18.8717 −0.754265
$$627$$ −0.429994 −0.0171723
$$628$$ −19.9316 −0.795359
$$629$$ 3.60861 0.143885
$$630$$ 0 0
$$631$$ 5.01448 0.199623 0.0998116 0.995006i $$-0.468176\pi$$
0.0998116 + 0.995006i $$0.468176\pi$$
$$632$$ −8.72499 −0.347061
$$633$$ 10.3767 0.412435
$$634$$ 15.1979 0.603587
$$635$$ 0 0
$$636$$ 9.34696 0.370631
$$637$$ 78.0747 3.09343
$$638$$ −0.189728 −0.00751142
$$639$$ 2.93762 0.116210
$$640$$ 0 0
$$641$$ 18.9133 0.747031 0.373515 0.927624i $$-0.378152\pi$$
0.373515 + 0.927624i $$0.378152\pi$$
$$642$$ −27.4788 −1.08450
$$643$$ 22.9344 0.904444 0.452222 0.891906i $$-0.350632\pi$$
0.452222 + 0.891906i $$0.350632\pi$$
$$644$$ −10.2863 −0.405336
$$645$$ 0 0
$$646$$ −22.4456 −0.883109
$$647$$ 10.4741 0.411779 0.205889 0.978575i $$-0.433991\pi$$
0.205889 + 0.978575i $$0.433991\pi$$
$$648$$ −6.61139 −0.259720
$$649$$ 0.610522 0.0239651
$$650$$ 0 0
$$651$$ −21.5447 −0.844405
$$652$$ 8.65087 0.338794
$$653$$ −44.7251 −1.75023 −0.875115 0.483916i $$-0.839214\pi$$
−0.875115 + 0.483916i $$0.839214\pi$$
$$654$$ 4.74745 0.185640
$$655$$ 0 0
$$656$$ −7.38299 −0.288257
$$657$$ −4.60111 −0.179506
$$658$$ −20.0000 −0.779681
$$659$$ 15.1056 0.588431 0.294215 0.955739i $$-0.404942\pi$$
0.294215 + 0.955739i $$0.404942\pi$$
$$660$$ 0 0
$$661$$ 34.9654 1.36000 0.679998 0.733214i $$-0.261981\pi$$
0.679998 + 0.733214i $$0.261981\pi$$
$$662$$ −23.0210 −0.894736
$$663$$ 29.1030 1.13027
$$664$$ −0.880231 −0.0341596
$$665$$ 0 0
$$666$$ −0.653743 −0.0253320
$$667$$ 9.25537 0.358369
$$668$$ −11.9640 −0.462901
$$669$$ −15.7094 −0.607361
$$670$$ 0 0
$$671$$ −0.192848 −0.00744483
$$672$$ 7.15650 0.276068
$$673$$ 35.2496 1.35877 0.679385 0.733782i $$-0.262246\pi$$
0.679385 + 0.733782i $$0.262246\pi$$
$$674$$ 34.0776 1.31262
$$675$$ 0 0
$$676$$ 14.7216 0.566217
$$677$$ −23.2835 −0.894858 −0.447429 0.894320i $$-0.647660\pi$$
−0.447429 + 0.894320i $$0.647660\pi$$
$$678$$ 6.68125 0.256592
$$679$$ 1.12321 0.0431049
$$680$$ 0 0
$$681$$ −17.2695 −0.661769
$$682$$ −0.135870 −0.00520274
$$683$$ −11.8131 −0.452017 −0.226008 0.974125i $$-0.572568\pi$$
−0.226008 + 0.974125i $$0.572568\pi$$
$$684$$ 4.06628 0.155478
$$685$$ 0 0
$$686$$ 36.5761 1.39648
$$687$$ −14.4060 −0.549624
$$688$$ −5.54789 −0.211511
$$689$$ 32.1286 1.22400
$$690$$ 0 0
$$691$$ 14.1858 0.539652 0.269826 0.962909i $$-0.413034\pi$$
0.269826 + 0.962909i $$0.413034\pi$$
$$692$$ −5.33508 −0.202809
$$693$$ −0.137849 −0.00523646
$$694$$ 13.8449 0.525547
$$695$$ 0 0
$$696$$ −6.43926 −0.244080
$$697$$ −26.6423 −1.00915
$$698$$ 11.3765 0.430607
$$699$$ −4.94552 −0.187057
$$700$$ 0 0
$$701$$ 19.2184 0.725870 0.362935 0.931815i $$-0.381775\pi$$
0.362935 + 0.931815i $$0.381775\pi$$
$$702$$ −29.4670 −1.11216
$$703$$ −6.22000 −0.234592
$$704$$ 0.0451320 0.00170097
$$705$$ 0 0
$$706$$ 19.2631 0.724976
$$707$$ −26.2468 −0.987111
$$708$$ 20.7207 0.778733
$$709$$ 4.52525 0.169949 0.0849746 0.996383i $$-0.472919\pi$$
0.0849746 + 0.996383i $$0.472919\pi$$
$$710$$ 0 0
$$711$$ 5.70390 0.213913
$$712$$ 9.97602 0.373867
$$713$$ 6.62805 0.248222
$$714$$ 25.8250 0.966478
$$715$$ 0 0
$$716$$ −5.27905 −0.197288
$$717$$ −27.3589 −1.02174
$$718$$ 14.6645 0.547273
$$719$$ 0.915477 0.0341415 0.0170708 0.999854i $$-0.494566\pi$$
0.0170708 + 0.999854i $$0.494566\pi$$
$$720$$ 0 0
$$721$$ −58.7110 −2.18651
$$722$$ 19.6884 0.732728
$$723$$ 40.8732 1.52009
$$724$$ 17.7448 0.659478
$$725$$ 0 0
$$726$$ −16.8461 −0.625218
$$727$$ −11.7429 −0.435521 −0.217760 0.976002i $$-0.569875\pi$$
−0.217760 + 0.976002i $$0.569875\pi$$
$$728$$ 24.5993 0.911710
$$729$$ 30.0400 1.11259
$$730$$ 0 0
$$731$$ −20.0202 −0.740473
$$732$$ −6.54515 −0.241916
$$733$$ 27.6776 1.02230 0.511148 0.859493i $$-0.329220\pi$$
0.511148 + 0.859493i $$0.329220\pi$$
$$734$$ 0.868349 0.0320514
$$735$$ 0 0
$$736$$ −2.20164 −0.0811534
$$737$$ 0.558514 0.0205731
$$738$$ 4.82658 0.177669
$$739$$ −6.52187 −0.239911 −0.119956 0.992779i $$-0.538275\pi$$
−0.119956 + 0.992779i $$0.538275\pi$$
$$740$$ 0 0
$$741$$ −50.1635 −1.84280
$$742$$ 28.5099 1.04663
$$743$$ 43.0091 1.57785 0.788925 0.614489i $$-0.210638\pi$$
0.788925 + 0.614489i $$0.210638\pi$$
$$744$$ −4.61135 −0.169060
$$745$$ 0 0
$$746$$ 32.4555 1.18828
$$747$$ 0.575445 0.0210544
$$748$$ 0.162864 0.00595489
$$749$$ −83.8152 −3.06254
$$750$$ 0 0
$$751$$ −29.7068 −1.08402 −0.542008 0.840373i $$-0.682336\pi$$
−0.542008 + 0.840373i $$0.682336\pi$$
$$752$$ −4.28072 −0.156102
$$753$$ −46.1902 −1.68327
$$754$$ −22.1339 −0.806069
$$755$$ 0 0
$$756$$ −26.1480 −0.950994
$$757$$ 32.7918 1.19184 0.595920 0.803044i $$-0.296788\pi$$
0.595920 + 0.803044i $$0.296788\pi$$
$$758$$ −30.2511 −1.09877
$$759$$ −0.152201 −0.00552455
$$760$$ 0 0
$$761$$ 22.9411 0.831616 0.415808 0.909452i $$-0.363499\pi$$
0.415808 + 0.909452i $$0.363499\pi$$
$$762$$ 27.0465 0.979790
$$763$$ 14.4806 0.524232
$$764$$ −3.11649 −0.112751
$$765$$ 0 0
$$766$$ 14.4935 0.523673
$$767$$ 71.2241 2.57175
$$768$$ 1.53175 0.0552723
$$769$$ 10.8123 0.389902 0.194951 0.980813i $$-0.437545\pi$$
0.194951 + 0.980813i $$0.437545\pi$$
$$770$$ 0 0
$$771$$ −4.83078 −0.173976
$$772$$ −8.78278 −0.316099
$$773$$ −37.9989 −1.36673 −0.683363 0.730079i $$-0.739483\pi$$
−0.683363 + 0.730079i $$0.739483\pi$$
$$774$$ 3.62689 0.130366
$$775$$ 0 0
$$776$$ 0.240408 0.00863014
$$777$$ 7.15650 0.256738
$$778$$ 36.9033 1.32305
$$779$$ 45.9222 1.64533
$$780$$ 0 0
$$781$$ −0.202802 −0.00725683
$$782$$ −7.94485 −0.284107
$$783$$ 23.5274 0.840801
$$784$$ 14.8286 0.529593
$$785$$ 0 0
$$786$$ −15.7927 −0.563309
$$787$$ 31.7730 1.13258 0.566292 0.824205i $$-0.308377\pi$$
0.566292 + 0.824205i $$0.308377\pi$$
$$788$$ 7.00278 0.249464
$$789$$ 21.4953 0.765252
$$790$$ 0 0
$$791$$ 20.3790 0.724594
$$792$$ −0.0295047 −0.00104840
$$793$$ −22.4979 −0.798923
$$794$$ 22.9365 0.813986
$$795$$ 0 0
$$796$$ 17.3202 0.613900
$$797$$ 9.02551 0.319700 0.159850 0.987141i $$-0.448899\pi$$
0.159850 + 0.987141i $$0.448899\pi$$
$$798$$ −44.5135 −1.57576
$$799$$ −15.4475 −0.546492
$$800$$ 0 0
$$801$$ −6.52175 −0.230435
$$802$$ 27.4332 0.968698
$$803$$ 0.317643 0.0112094
$$804$$ 18.9556 0.668513
$$805$$ 0 0
$$806$$ −15.8508 −0.558319
$$807$$ −46.1450 −1.62438
$$808$$ −5.61775 −0.197632
$$809$$ 29.7376 1.04552 0.522759 0.852481i $$-0.324903\pi$$
0.522759 + 0.852481i $$0.324903\pi$$
$$810$$ 0 0
$$811$$ 15.7650 0.553585 0.276793 0.960930i $$-0.410729\pi$$
0.276793 + 0.960930i $$0.410729\pi$$
$$812$$ −19.6409 −0.689260
$$813$$ 22.7928 0.799378
$$814$$ 0.0451320 0.00158187
$$815$$ 0 0
$$816$$ 5.52749 0.193501
$$817$$ 34.5079 1.20728
$$818$$ −21.3920 −0.747954
$$819$$ −16.0816 −0.561937
$$820$$ 0 0
$$821$$ −4.27798 −0.149303 −0.0746513 0.997210i $$-0.523784\pi$$
−0.0746513 + 0.997210i $$0.523784\pi$$
$$822$$ 4.93014 0.171958
$$823$$ −13.8985 −0.484471 −0.242236 0.970217i $$-0.577881\pi$$
−0.242236 + 0.970217i $$0.577881\pi$$
$$824$$ −12.5663 −0.437766
$$825$$ 0 0
$$826$$ 63.2019 2.19908
$$827$$ −30.4944 −1.06039 −0.530197 0.847875i $$-0.677882\pi$$
−0.530197 + 0.847875i $$0.677882\pi$$
$$828$$ 1.43930 0.0500192
$$829$$ 0.493020 0.0171233 0.00856165 0.999963i $$-0.497275\pi$$
0.00856165 + 0.999963i $$0.497275\pi$$
$$830$$ 0 0
$$831$$ 0.225593 0.00782574
$$832$$ 5.26514 0.182536
$$833$$ 53.5107 1.85404
$$834$$ 32.0749 1.11066
$$835$$ 0 0
$$836$$ −0.280721 −0.00970894
$$837$$ 16.8487 0.582376
$$838$$ 23.0891 0.797598
$$839$$ −27.9308 −0.964278 −0.482139 0.876095i $$-0.660140\pi$$
−0.482139 + 0.876095i $$0.660140\pi$$
$$840$$ 0 0
$$841$$ −11.3276 −0.390606
$$842$$ −1.83757 −0.0633268
$$843$$ −39.0975 −1.34659
$$844$$ 6.77438 0.233184
$$845$$ 0 0
$$846$$ 2.79849 0.0962141
$$847$$ −51.3837 −1.76556
$$848$$ 6.10215 0.209549
$$849$$ 0.0861901 0.00295804
$$850$$ 0 0
$$851$$ −2.20164 −0.0754711
$$852$$ −6.88297 −0.235807
$$853$$ 31.4029 1.07521 0.537607 0.843195i $$-0.319328\pi$$
0.537607 + 0.843195i $$0.319328\pi$$
$$854$$ −19.9639 −0.683150
$$855$$ 0 0
$$856$$ −17.9395 −0.613158
$$857$$ 39.7806 1.35888 0.679441 0.733731i $$-0.262223\pi$$
0.679441 + 0.733731i $$0.262223\pi$$
$$858$$ 0.363983 0.0124262
$$859$$ 41.8673 1.42849 0.714247 0.699894i $$-0.246769\pi$$
0.714247 + 0.699894i $$0.246769\pi$$
$$860$$ 0 0
$$861$$ −52.8364 −1.80066
$$862$$ −15.9796 −0.544267
$$863$$ −8.52387 −0.290156 −0.145078 0.989420i $$-0.546343\pi$$
−0.145078 + 0.989420i $$0.546343\pi$$
$$864$$ −5.59662 −0.190401
$$865$$ 0 0
$$866$$ −25.2525 −0.858116
$$867$$ −6.09319 −0.206936
$$868$$ −14.0654 −0.477412
$$869$$ −0.393776 −0.0133579
$$870$$ 0 0
$$871$$ 65.1568 2.20775
$$872$$ 3.09936 0.104958
$$873$$ −0.157165 −0.00531922
$$874$$ 13.6942 0.463213
$$875$$ 0 0
$$876$$ 10.7806 0.364243
$$877$$ −37.1163 −1.25333 −0.626664 0.779289i $$-0.715580\pi$$
−0.626664 + 0.779289i $$0.715580\pi$$
$$878$$ −9.30792 −0.314127
$$879$$ 20.9901 0.707977
$$880$$ 0 0
$$881$$ 29.8633 1.00612 0.503060 0.864252i $$-0.332208\pi$$
0.503060 + 0.864252i $$0.332208\pi$$
$$882$$ −9.69410 −0.326417
$$883$$ −9.98952 −0.336174 −0.168087 0.985772i $$-0.553759\pi$$
−0.168087 + 0.985772i $$0.553759\pi$$
$$884$$ 18.9998 0.639033
$$885$$ 0 0
$$886$$ −9.52445 −0.319980
$$887$$ 23.2703 0.781340 0.390670 0.920531i $$-0.372243\pi$$
0.390670 + 0.920531i $$0.372243\pi$$
$$888$$ 1.53175 0.0514022
$$889$$ 82.4965 2.76684
$$890$$ 0 0
$$891$$ −0.298385 −0.00999628
$$892$$ −10.2559 −0.343392
$$893$$ 26.6261 0.891008
$$894$$ −33.6728 −1.12619
$$895$$ 0 0
$$896$$ 4.67211 0.156084
$$897$$ −17.7559 −0.592852
$$898$$ 5.43278 0.181294
$$899$$ 12.6558 0.422094
$$900$$ 0 0
$$901$$ 22.0203 0.733602
$$902$$ −0.333209 −0.0110946
$$903$$ −39.7035 −1.32125
$$904$$ 4.36184 0.145073
$$905$$ 0 0
$$906$$ 21.7320 0.721998
$$907$$ −34.0053 −1.12913 −0.564564 0.825389i $$-0.690956\pi$$
−0.564564 + 0.825389i $$0.690956\pi$$
$$908$$ −11.2744 −0.374153
$$909$$ 3.67256 0.121811
$$910$$ 0 0
$$911$$ −29.2297 −0.968424 −0.484212 0.874951i $$-0.660894\pi$$
−0.484212 + 0.874951i $$0.660894\pi$$
$$912$$ −9.52749 −0.315487
$$913$$ −0.0397266 −0.00131476
$$914$$ 7.54789 0.249662
$$915$$ 0 0
$$916$$ −9.40494 −0.310748
$$917$$ −48.1707 −1.59074
$$918$$ −20.1960 −0.666568
$$919$$ 43.0238 1.41923 0.709613 0.704592i $$-0.248870\pi$$
0.709613 + 0.704592i $$0.248870\pi$$
$$920$$ 0 0
$$921$$ −26.9317 −0.887430
$$922$$ −35.4323 −1.16690
$$923$$ −23.6591 −0.778748
$$924$$ 0.322987 0.0106255
$$925$$ 0 0
$$926$$ 20.9124 0.687225
$$927$$ 8.21510 0.269819
$$928$$ −4.20386 −0.137998
$$929$$ 10.7786 0.353635 0.176817 0.984244i $$-0.443420\pi$$
0.176817 + 0.984244i $$0.443420\pi$$
$$930$$ 0 0
$$931$$ −92.2340 −3.02285
$$932$$ −3.22867 −0.105759
$$933$$ 0.540912 0.0177087
$$934$$ 6.62523 0.216784
$$935$$ 0 0
$$936$$ −3.44204 −0.112507
$$937$$ 36.5755 1.19487 0.597435 0.801918i $$-0.296187\pi$$
0.597435 + 0.801918i $$0.296187\pi$$
$$938$$ 57.8180 1.88782
$$939$$ −28.9067 −0.943336
$$940$$ 0 0
$$941$$ 58.9579 1.92197 0.960986 0.276596i $$-0.0892064\pi$$
0.960986 + 0.276596i $$0.0892064\pi$$
$$942$$ −30.5303 −0.994730
$$943$$ 16.2547 0.529325
$$944$$ 13.5275 0.440282
$$945$$ 0 0
$$946$$ −0.250387 −0.00814079
$$947$$ 9.63990 0.313255 0.156627 0.987658i $$-0.449938\pi$$
0.156627 + 0.987658i $$0.449938\pi$$
$$948$$ −13.3645 −0.434059
$$949$$ 37.0566 1.20291
$$950$$ 0 0
$$951$$ 23.2794 0.754887
$$952$$ 16.8598 0.546430
$$953$$ 31.4809 1.01977 0.509884 0.860243i $$-0.329688\pi$$
0.509884 + 0.860243i $$0.329688\pi$$
$$954$$ −3.98923 −0.129156
$$955$$ 0 0
$$956$$ −17.8612 −0.577672
$$957$$ −0.290616 −0.00939430
$$958$$ −29.4581 −0.951747
$$959$$ 15.0378 0.485596
$$960$$ 0 0
$$961$$ −21.9368 −0.707639
$$962$$ 5.26514 0.169755
$$963$$ 11.7278 0.377923
$$964$$ 26.6840 0.859434
$$965$$ 0 0
$$966$$ −15.7560 −0.506941
$$967$$ 25.5439 0.821438 0.410719 0.911762i $$-0.365278\pi$$
0.410719 + 0.911762i $$0.365278\pi$$
$$968$$ −10.9980 −0.353488
$$969$$ −34.3810 −1.10448
$$970$$ 0 0
$$971$$ 43.7224 1.40312 0.701559 0.712611i $$-0.252488\pi$$
0.701559 + 0.712611i $$0.252488\pi$$
$$972$$ 6.66286 0.213711
$$973$$ 97.8341 3.13642
$$974$$ 8.80654 0.282180
$$975$$ 0 0
$$976$$ −4.27299 −0.136775
$$977$$ −44.6598 −1.42879 −0.714396 0.699741i $$-0.753299\pi$$
−0.714396 + 0.699741i $$0.753299\pi$$
$$978$$ 13.2510 0.423719
$$979$$ 0.450237 0.0143896
$$980$$ 0 0
$$981$$ −2.02619 −0.0646912
$$982$$ 5.14258 0.164106
$$983$$ 9.17840 0.292745 0.146373 0.989230i $$-0.453240\pi$$
0.146373 + 0.989230i $$0.453240\pi$$
$$984$$ −11.3089 −0.360514
$$985$$ 0 0
$$986$$ −15.1701 −0.483114
$$987$$ −30.6350 −0.975123
$$988$$ −32.7492 −1.04189
$$989$$ 12.2144 0.388397
$$990$$ 0 0
$$991$$ 16.1246 0.512216 0.256108 0.966648i $$-0.417560\pi$$
0.256108 + 0.966648i $$0.417560\pi$$
$$992$$ −3.01051 −0.0955839
$$993$$ −35.2624 −1.11902
$$994$$ −20.9943 −0.665899
$$995$$ 0 0
$$996$$ −1.34829 −0.0427223
$$997$$ 8.99979 0.285026 0.142513 0.989793i $$-0.454482\pi$$
0.142513 + 0.989793i $$0.454482\pi$$
$$998$$ −12.9692 −0.410532
$$999$$ −5.59662 −0.177069
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1850.2.a.be.1.4 5
5.2 odd 4 370.2.b.d.149.7 yes 10
5.3 odd 4 370.2.b.d.149.4 10
5.4 even 2 1850.2.a.bd.1.2 5
15.2 even 4 3330.2.d.p.1999.1 10
15.8 even 4 3330.2.d.p.1999.6 10

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.4 10 5.3 odd 4
370.2.b.d.149.7 yes 10 5.2 odd 4
1850.2.a.bd.1.2 5 5.4 even 2
1850.2.a.be.1.4 5 1.1 even 1 trivial
3330.2.d.p.1999.1 10 15.2 even 4
3330.2.d.p.1999.6 10 15.8 even 4